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Data-driven design approach to hierarchical hybrid structures with multiple lattice configurations

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Abstract

This work presents a data-driven design approach to hierarchical hybrid structures with multiple lattice configurations. Two design variables are considered for each lattice substructure, one discrete variable indicating the configuration type and the other continuous density variable determining the geometrical feature size. For each lattice configuration, a series of similar lattice substructures are sampled by varying the density variable and a corresponding data-driven interpolation model is built for an explicit representation of the constitutive behavior. To reduce the model complexity, substructuring by means of static condensation is performed on the sampled lattice substructures. To achieve hybrid structure with multiple lattice configurations, a multi-material interpolation model is adopted by synthesizing the data-driven interpolation models and the discrete lattice configuration variables. The proposed approach has proved capable of generating hierarchically strongly coupled designs, which therefore allows for direct manufacturing with no post-processing requirement as required for homogenization-based designs due to the assumption on scales separation.

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References

  • Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202

    Article  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MathSciNet  Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654

    MATH  Google Scholar 

  • Berger JB, Wadley HNG, McMeeking RM (2017) Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543:533–537

    Article  Google Scholar 

  • Da D, Yvonnet J, Xia L, Le MV, Li GY (2018) Topology optimization of periodic lattice structures taking into account strain gradient. Comput Struct 210:28–40

    Article  Google Scholar 

  • Fu JJ, Xia L, Gao L, Xiao M, Li H (2019) Topology optimization of periodic structures with substructuring. ASME J Mech Des 141(7):071403

    Article  Google Scholar 

  • Gao T, Zhang WH (2011) A mass constraint formulation for structural topology optimization with multiphase materials. Int J Numer Methods Eng 88(8):774–796

    Article  Google Scholar 

  • Gibiansky LV, Sigmund O (2000) Multiphase composites with extremal bulk modulus. J Mech Phys Solids 48(3):461–498

    Article  MathSciNet  Google Scholar 

  • Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43(3):393–401

    Article  MathSciNet  Google Scholar 

  • Hvejsel C, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825

    Article  Google Scholar 

  • Latture RM, Begley MR, Zok FW (2018) Design and mechanical properties of elastically isotropic trusses. J Mater Res 33:249–263

    Article  Google Scholar 

  • Long K, Wang X, Gu XG (2018) Local optimum in multi-material topology optimization and solution by reciprocal variables. Struct Multidiscip Optim 57(3):1283–1295

    Article  MathSciNet  Google Scholar 

  • Meng L, Zhang WH, Quan D, Shi G, Tang L, Hou Y, Breitkopf P, Zhu JH, Gao T (2019) From topology optimization design to additive manufacturing: today’s success and tomorrow’s roadmap. Arch Computat Methods Eng:1–26. https://doi.org/10.1007/s11831-019-09331-1

  • Rodrigues H, Guedes JM, Bendsøe MP (2002) Hierarchical optimization of material and structure. Struct Multidiscip Optim 24(1):1–10

    Article  Google Scholar 

  • Sanders ED, Aguiló MA, Paulino GH (2018) Multi-material continuum topology optimization with arbitrary volume and mass constraints. Comput Methods Appl Mech Eng 340:798–823

    Article  MathSciNet  Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches-a comparative review. Struct Multidiscip Optim 48(6):1031–1055

    Article  MathSciNet  Google Scholar 

  • Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067

    Article  MathSciNet  Google Scholar 

  • Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027

    Article  Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    Article  MathSciNet  Google Scholar 

  • Thomsen J (1992) Topology optimization of structures composed of one or two materials. Struct Optim 5(1–2):108–115

    Article  Google Scholar 

  • Wang MY, Wang XM (2004) “Color” level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6):469–496

    Article  MathSciNet  Google Scholar 

  • Wang FW, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784

    Article  Google Scholar 

  • Wang Y, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586

    Article  MathSciNet  Google Scholar 

  • Wang YJ, Xu H, Pasini D (2017) Multiscale isogeometric topology optimization for lattice materials. Comput Methods Appl Mech Eng 316:568–585

    Article  MathSciNet  Google Scholar 

  • Wang C, Zhu JH, Zhang WH, Li SY, Kong J (2018a) Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructure. Struct Multidiscip Optim 58(1):35–50

    Article  MathSciNet  Google Scholar 

  • Wang Y, Zhang L, Daynes S, Zhang H, Feih S, Wang MY (2018b) Design of graded lattice structure with optimized mesostructures for additive manufacturing. Mater Des 142:114–123

    Article  Google Scholar 

  • Wang Y, Groen JP, Sigmund O (2019) Simple optimal lattice structures for arbitrary loadings. Extreme Mechanics Letters 29:100447

    Article  Google Scholar 

  • Wei P, Ma H, Wang MY (2014) The stiffness spreading method for layout optimization of truss structures. Struct Multidiscip Optim 49(2):667–682

    Article  MathSciNet  Google Scholar 

  • Wu ZJ, Xia L, Wang ST, Shi TL (2019) Topology optimization of hierarchical lattice structures with substructuring. Comput Methods Appl Mech Eng 345:602–617

    Article  MathSciNet  Google Scholar 

  • Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542

    Article  Google Scholar 

  • Xia L, Breitkopf P (2015) Multiscale structural topology optimization with an approximate constitutive model for local material microstructure. Comput Methods Appl Mech Eng 286:147–167

    Article  MathSciNet  Google Scholar 

  • Xia L, Breitkopf P (2017) Recent advances on topology optimization of multiscale nonlinear structures. Arch Computat Methods Eng 24(2):227–249

    Article  MathSciNet  Google Scholar 

  • Xia L, Xia Q, Huang X, Xie Y (2018) Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch Computat Methods Eng 25(2):437–478

    Article  MathSciNet  Google Scholar 

  • Xu MM, Xia L, Wang ST, Liu HL, Xie XD (2019) An isogeometric approach to topology optimization of spatially graded hierarchical structures. Compos Struct 225(1):111171

    Article  Google Scholar 

  • Zhang WH, Sun SP (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68(9):993–1011

    Article  Google Scholar 

  • Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336

    Article  Google Scholar 

  • Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Computat Methods Eng 23(4):595–622

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China (51705165, 11972166, 51790171, 51975227) and the Fundamental Research Funds for the Central Universities (2019kfyXKJC044).

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Correspondence to Liang Xia.

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The authors declare that they have no conflict of interest.

Replication of results

MATLAB codes for the cantilever example presented in Figs. 6c and 7c are available as supplementary material. Design cases of the half MBB beam can be easily replicated with the provided codes by changing the boundary conditions.

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Responsible Editor: Emilio Carlos Nelli Silva

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Liu, Z., Xia, L., Xia, Q. et al. Data-driven design approach to hierarchical hybrid structures with multiple lattice configurations. Struct Multidisc Optim 61, 2227–2235 (2020). https://doi.org/10.1007/s00158-020-02497-4

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  • DOI: https://doi.org/10.1007/s00158-020-02497-4

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